Why does the water keep escaping in my fluid simulation?
As others have mentioned, it's the resolution that's key here. However, I see that you've baked at two separate resolutions: 120 for final and 45 for preview.

You can see from the render view that the "final" quality bake has less "escaping" water. To preview that in your viewport without rendering, change the "Viewport display" dropdown to "Final." The downside of this is that it may slow down your viewport.

If you want even better quality, you could either:


increase the resolution, or:
Alt-D to linked-duplicate the cup object, then scale it down a tad (maybe 95%) and position it inside the real cup. Turn off the viewport and render visibility. Set its parent to the "real" cup, but mark both as collision targets (Physics &rarr; Fluid &rarr; Obstacle). This will essentially shrink the bounds of your cup and may or may not completely solve the problem.